The $ L ^{ 1 } $ -Potts Functional for Robust Jump-Sparse Reconstruction

We investigate the nonsmooth and nonconvex $ L ^{ 1 } $ -Potts functional in discrete and continuous time. We show Γ-convergence of discrete $ L ^{ 1 } $ -Potts functionals toward their continuous counterpart and obtain a convergence statement for the corresponding minimizers as the discretization gets finer. For the discrete $ L ^{ 1 } $ -Potts problem, we introduce an $ O(n ^{ 2 } $ ) time and O(n) space algorithm to compute an exact minimizer. We apply $ L ^{ 1 } $ -Potts minimization to the problem of recovering piecewise constant signals from noisy measurements ƒ. It turns out that the $ L ^{ 1 } $ -Potts functional has a quite interesting blind deconvolution property. In fact, we show that mildly blurred jump-sparse signals are reconstructed by minimizing the $ L ^{ 1 } $ -Potts functional. Furthermore, for strongly blurred signals and a known blurring operator, we derive an iterative reconstruction algorithm.


Published in:
SIAM Journal on Numerical Analysis, 53, 1, 644–673
Year:
2015
Publisher:
SIAM
Laboratories:




 Record created 2015-07-28, last modified 2018-03-17

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